Coefficient of determination
Measure the proportion of response variation explained by a fitted model.
Coefficient of determination is one of 3 regression formulas in the statistics section of this library, and it is used at high school · ap · university level.
Why coefficient of determination works
The total sum of squares measures how much the response varies around its own average, which is what you would face with no model at all. The residual sum of squares measures what is left after fitting. Their ratio is the share still unexplained, so subtracting from 1 gives the share the model accounts for.
What each symbol means
$SS_{\mathrm{res}}$ is residual sum of squares and $SS_{\mathrm{tot}}$ total sum of squares.
Coefficient of determination: when it holds
Interpret within the fitted data and model; a large $R^2$ does not prove causation or a correct model.
When it stops applying
It only reports fit to the data in front of you, and it can never go down when a variable is added, even a column of pure noise. A high value therefore says nothing about whether the model is correct, honest, or able to predict a case it has not seen.
Coefficient of determination: a worked example
If residual variation is $20$ of total variation $100$, then $R^2=0.80$.
The mistake to avoid
What people do: Reading a value of 0.80 as 80 percent of predictions being correct.
Why it goes wrong: It is a share of variation, not a hit rate. A model can explain 80 percent of the variation and still miss every single observation by a noticeable amount.
Do this instead: Say it as the percentage of variation in the response explained by the model, and quote a typical residual size if you want to describe accuracy.
Coefficient of determination: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $SS_{\mathrm{res}}$ is residual sum of squares and $SS_{\mathrm{tot}}$ total sum of squares.
- Check the conditions before substituting. Interpret within the fitted data and model; a large $R^2$ does not prove causation or a correct model.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most statistics slips.
Where this formula fits
- Subject
- Statistics formulas — 21 entries in this library
- Topic
- Regression
- Level
- High school · AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where coefficient of determination comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Linear Regression — the lesson behind this formula: fit and interpret a linear relationship honestly.
- Statistics Calculator — check your substitution and the value it produces.
- Study statistics — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 21 statistics formulas — the full grouped reference, or the complete formula library.
Questions about coefficient of determination
Is it always the correlation squared?
For simple linear regression with an intercept, yes. A correlation of 0.8 gives 0.64. With more than one predictor the relationship no longer holds in that simple form.
Can it come out negative?
Yes, when a model is not fitted by least squares on the same data, such as a forced intercept or a model scored on new observations. A negative value means the model does worse than just predicting the mean.
What counts as a good value?
There is no universal threshold. Laboratory measurements might demand above 0.95, while a model of human behaviour explaining 0.30 of the variation can be a genuinely useful finding.
Does it tell me whether the slope is statistically significant?
No, that is a separate question answered by a t test on the slope. With a very large sample a tiny value can still come with a highly significant slope.